| L(s) = 1 | + (−0.733 − 0.576i)2-s + (−0.358 + 0.503i)3-s + (−0.266 − 1.09i)4-s + (−3.83 + 0.738i)5-s + (0.553 − 0.162i)6-s + (1.47 + 2.19i)7-s + (−1.21 + 2.65i)8-s + (0.856 + 2.47i)9-s + (3.23 + 1.66i)10-s + (−1.69 + 1.32i)11-s + (0.648 + 0.259i)12-s + (0.00277 − 0.00178i)13-s + (0.190 − 2.46i)14-s + (1.00 − 2.19i)15-s + (0.413 − 0.213i)16-s + (−5.42 − 5.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 − 0.407i)2-s + (−0.206 + 0.290i)3-s + (−0.133 − 0.548i)4-s + (−1.71 + 0.330i)5-s + (0.225 − 0.0663i)6-s + (0.555 + 0.831i)7-s + (−0.428 + 0.939i)8-s + (0.285 + 0.824i)9-s + (1.02 + 0.527i)10-s + (−0.509 + 0.400i)11-s + (0.187 + 0.0748i)12-s + (0.000769 − 0.000494i)13-s + (0.0508 − 0.657i)14-s + (0.258 − 0.566i)15-s + (0.103 − 0.0532i)16-s + (−1.31 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.173277 + 0.254431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.173277 + 0.254431i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.47 - 2.19i)T \) |
| 23 | \( 1 + (-4.79 + 0.0906i)T \) |
| good | 2 | \( 1 + (0.733 + 0.576i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (0.358 - 0.503i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (3.83 - 0.738i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (1.69 - 1.32i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.00277 + 0.00178i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (5.42 + 5.17i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (4.04 - 3.85i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (4.90 - 1.44i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.62 - 0.250i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.268 - 0.775i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-4.66 - 5.38i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.84 + 6.23i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-3.03 + 5.26i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.306 - 6.44i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-9.04 - 4.66i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (0.0837 + 0.117i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-0.798 + 0.319i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.271 - 1.88i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.49 - 10.3i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.0546 + 1.14i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (1.69 - 1.95i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (7.39 + 0.706i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (1.92 + 2.22i)T + (-13.8 + 96.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.94869705986565790551552352399, −11.70145790610223529868135615004, −11.16587135851766500197995427105, −10.47453128660293513631044991913, −9.065886262889047908614644283420, −8.196124430712571522079362489155, −7.15273791700332019933832459098, −5.30096507738647818620828404076, −4.39362141745942351491715644754, −2.38648327737454390299589687418,
0.35039510679551985364481709771, 3.68896567736420990869254215249, 4.43232667440606070561524626951, 6.67136658854521105781803568530, 7.44489492464727499995366641636, 8.294033588109313716220534739465, 9.037299714390330856646096414272, 10.87927223491952469937680909421, 11.45267953907139225602925180312, 12.77059206983852914085892790169
